Visual models of identities

The correct answer is A. The expression \(a^2+2ab+b^2\) is the expansion of \((a+b)^2\), so the complete shape must be a square with side \(a+b\). Divide each side into parts of lengths \(a\) and \(b\). This creates one \(a\)-by-\(a\) square of area \(a^2\), one \(b\)-by-\(b\) square of area \(b^2\), and two \(a\)-by-\(b\) rectangles, each of area \(ab\). Their sum is exactly \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B has only one \(ab\) rectangle and does not form the stated complete square. Option C does not ensure the required areas or the terms \(a^2\), \(b^2\), and \(2ab\). Option D describes disconnected rectangles and omits both square terms. The key clue is the coefficient 2: it requires two equal cross-rectangles.

The correct answer is B: \(c-6\). Compare the expression with the identity \((a-b)^2=a^2-2ab+b^2\). The first term \(c^2\) shows that \(a=c\). The constant term 36 is \(6^2\), so the fixed part is 6. The middle term should then be \(-2(c)(6)=-12c\), exactly as given. Therefore, \(c^2-12c+36=(c-6)^2\), and the side of the square is \(c-6\). Option A gives \((c+6)^2=c^2+12c+36\), with the wrong sign in the middle. Option C gives \((c-12)^2=c^2-24c+144\), so both the middle and constant terms are wrong. Option D gives \((c+12)^2=c^2+24c+144\), which is also incorrect. The negative middle term is the visual clue that the fixed length is subtracted. Also, the square root of the constant term gives the magnitude 6, not 12.

The correct answer is D: \(9\). In the area model, separate the side \(4x+3\) into a variable part \(4x\) and a constant part 3. The small corner is formed where the constant part meets itself, so its two side lengths are both 3. Its area is therefore \(3\times3=3^2=9\). Expanding the whole square confirms this: \((4x+3)^2=(4x)^2+2(4x)(3)+3^2=16x^2+24x+9\). The final constant term 9 is exactly the area of the fixed corner. Option A is the area of the large variable square with side \(4x\). Option B is not the correct combined cross-term; one rectangle has area \(12x\), while two together give \(24x\). Option C is the area of both cross-rectangles. Thus none of those terms is the small corner. A useful cue is that the corner made from the constant part contains no variable, so its area must be a constant number.

Answer: A, two rectangles with side lengths \(a\) and \(b\). Imagine a large square whose side is split into parts \(a\) and \(b\) in both directions. The four regions are an \(a\times a\) square with area \(a^2\), two \(a imes b\) rectangles each with area \(ab\), and a \(b imes b\) square with area \(b^2\). The two equal rectangles together have area \(ab+ab=2ab\). Therefore option A correctly represents the mixed term. B represents only \(a^2\), and C represents only \(b^2\). D is not the small corner region; a square of side \(a+b\) would be the entire large square, not one part. Memory cue: the coefficient 2 appears because there are two identical cross-rectangles, not because one rectangle has area \(2ab\).

Answer: A, \((a+b)^2=a^2+2ab+b^2\). Add the four displayed areas: \(a^2+ab+ab+b^2=a^2+2ab+b^2\). The two rectangles produce the positive middle term because both have area \(ab\). The complete outer square has side length \(a+b\), since its side is made from consecutive lengths \(a\) and \(b\); therefore its total area is \((a+b)^2\). This proves option A. B would require the two rectangles to be negative or removed, which is not the model shown. C represents a product of conjugate expressions and has only the two square areas with opposite signs, not two positive rectangles. D incorrectly omits the cross-term \(2ab\). Memory cue: two positive \(ab\) rectangles mean “plus twice ab”; two negative cross-parts would indicate the square-of-a-difference identity.

The direct answer is B: a square of side (x+6). A rectangle becomes a square when its two side lengths are equal. Here both dimensions are exactly the same, namely x+6. Therefore, the model is a square whose side is x+6, and its area would be (x+6)² if an area were required. Option A is wrong because x+12 is obtained by adding the two side expressions, but the side of a square is not found by adding the rectangle’s sides. Option C changes the plus sign to a minus sign without justification. Option D multiplies x by 6, which is not the given side length. A useful check is to choose x=2: both original sides become 8, so the figure is a square of side 8, not 14, 4, or 12. Memory cue: equal length and width means square.

The direct answer is B: 4b². In the visual model, the side a+2b is split into lengths a and 2b. The small square is the part whose side is 2b. The area of any square is side×side, so its area is (2b)²=2b×2b=4b². Option A forgets one factor of 2 and therefore gives only half the correct coefficient. Option C, 2ab, is the area of one mixed rectangle with sides a and 2b, not the area of a square. Option D, a², is the area of the other square, whose side is a. The complete identity is (a+2b)²=a²+4ab+4b², and the last term represents this small square. A quick check with a=3 and b=2 gives small-square side 4 and area 16; 4b²=4×4=16. Memory cue: for a square, square the complete side expression of that part.

The correct answer is A: 18. In a rectangle formed from (x−2) and (x−9), the constant corner comes from multiplying the two constant parts, −2 and −9. Thus (−2)(−9)=18. The two negative signs cancel because the product of two negative numbers is positive. The complete expansion would be x²−9x−2x+18=x²−11x+18, and the constant term is the corner 18. Option B has the correct magnitude but the wrong sign. Options C and D come from adding or subtracting the constants rather than multiplying them. They do not represent the corner area. Memory cue: the corner of a binomial rectangle is always the product of the two constant terms; check the sign before choosing.

The correct answer is B. Draw a square with side a+b and divide each side into parts a and b. The a-by-a corner forms one square of area a². The b-by-b corner forms one square of area b². The two remaining parts have dimensions a by b and b by a; each has area ab, so together they are two ab rectangles. The full area is therefore a²+ab+ab+b²=a²+2ab+b². Option A misses one cross-rectangle. Option C has two a² squares even though there is only one a-by-a region. Option D incorrectly counts two b² squares. Memory cue: in (a+b)², the two middle pieces are equal rectangles, which is why the middle term is 2ab.

The correct answer is A: (a+b)²=a²+2ab+b². The whole figure is a square whose side is a+b, so its area is (a+b)². Dividing both directions into lengths a and b creates one a² square, one b² square, and two rectangles of area ab. Adding the four parts gives a²+ab+ab+b²=a²+2ab+b². Option B describes the square of a difference and would require negative or subtractive cross-regions. Option C is the difference-of-squares identity, involving two different squares rather than this single divided square. Option D counts only one ab rectangle and is incomplete. Memory cue: a plus sign in both side lengths produces two positive cross-terms.

The correct answer is A: a²+2a+1. The area of any square is side × side. Here, each side is (a+1), so the total area is (a+1)(a+1)=(a+1)². Using the square identity (x+y)²=x²+2xy+y², with x=a and y=1, we get a²+2(a)(1)+1²=a²+2a+1. In the area model, the four parts are an a×a square of area a², two rectangles each of area a×1=a, and a 1×1 square of area 1. Therefore their sum is a²+a+a+1=a²+2a+1. Option B misses one a-sized rectangle, so its middle term is too small. Option C leaves out both mixed rectangles. Option D has the wrong leading term and does not represent the four pieces. Memory cue: for (a+1)², remember square, twice the product, square: a²+2a+1.

The correct answer is C: p²−16. The area of a rectangle is length multiplied by breadth, so the required area is (p+4)(p−4). This is the difference-of-squares pattern (x+y)(x−y)=x²−y². Taking x=p and y=4 gives p²−4²=p²−16. If we multiply directly, p·p=p², p·(−4)=−4p, 4·p=4p, and 4·(−4)=−16. The two middle terms cancel: −4p+4p=0. Option A has the wrong sign for the constant term. Option B is the expansion of (p+4)², not a product of a sum and a difference. Option D is the expansion of (p−4)², so it has an unnecessary negative middle term. Memory cue: equal numbers added and subtracted give a difference of squares, and the middle terms disappear.

The correct answer is D: 36. A square with side y+6 can be divided into a y×y square, two y×6 rectangles, and a small 6×6 corner square. The corner square uses the constant part of the side, which is 6 on both dimensions. Therefore its area is 6×6=36. The complete identity is (y+6)²=y²+12y+36. Option A is the area of the large y×y part, not the corner. Option B is the area of only one rectangle, y×6=6y. Option C is the combined area of the two rectangles, 6y+6y=12y. Option D correctly represents the remaining 6×6 square. Memory cue: in (variable+constant)², the small corner is the square of the constant: 6²=36.

The correct answer is A: 4m. Split each side of the square into lengths m and 2. The area model for (m+2)² then contains an m×m square, two equal rectangles, and a 2×2 square. Each rectangle has dimensions m and 2, so its area is m×2=2m. Since there are two such rectangles, their total area is 2m+2m=4m. Option B is the area of only one rectangle. Option C is the area of the m×m square, not the two mixed rectangles. Option D is the area of the 2×2 corner square. The complete expansion confirms the result: (m+2)²=m²+4m+4. Memory cue: the two equal cross-pieces together give the middle term 2×m×2=4m.

The correct answer is C: 8r. Use the rectangle model by multiplying each part of the first expression by each part of the second: r×r=r², r×5=5r, 3×r=3r, and 3×5=15. The two terms containing r are 5r and 3r. They are like terms because both have the same variable r to the same power, so they can be added: 5r+3r=8r. The complete expansion is r²+8r+15. Option A is only one mixed rectangle. Option B is the other mixed rectangle. Option D is the constant term obtained from 3×5, so it has no r factor. Memory cue: for (r+a)(r+b), the combined r-term is (a+b)r; here (3+5)r=8r.

The correct answer is A: 25. Squaring the side gives (n−5)². Using (x−y)²=x²−2xy+y², with x=n and y=5, we obtain n²−10n+25. The last term is the square of the constant part: (−5)²=25. In a visual model, the n×n square gives n², the two signed mixed parts together give −10n, and the small 5×5 square gives 25. Option B incorrectly keeps a negative sign after squaring; multiplying −5 by −5 produces positive 25. Option C is only part of a mixed-term calculation. Option D is the middle term, not the final constant term. Memory cue: the constant corner is always the square of 5, and a negative number squared is positive.

The correct answer is C: u+5. The total area is u²+10u+25. The first term u² indicates a square with side u. The last term 25 is 5², so the small corner square has side 5. The two mixed rectangles should then have total area 2×u×5=10u, exactly matching the middle part. Hence the complete square has side u+5, and (u+5)²=u²+10u+25. Option A treats 10 as a side even though it is not the square root of the constant area. Option B similarly uses 25 instead of its square root. Option D would produce (u−5)²=u²−10u+25, with the wrong sign for the middle term. Memory cue: for a²+2ab+b², identify a from the first term and b from the last term, then verify the middle term.

The correct answer is B: 12. In a visual area model, split one side into lengths x and 2, and the other side into lengths x and 6. This creates four smaller rectangles. The small corner formed by the two numerical pieces has side lengths 2 and 6. Therefore, its area is length × breadth = 2 × 6 = 12. Option A, 8, is not the product of the two corner lengths; it is only their sum. Option B is correct because it uses both dimensions of the small corner. Option C, 2x, is the area of the rectangle whose sides are 2 and x. Option D, 6x, is the area of the rectangle whose sides are 6 and x. Thus, the complete expansion is (x+2)(x+6)=x²+8x+12. Memory cue: in an area model, multiply the two side labels surrounding the particular region you are asked about.

The correct answer is A: rectangles with sides a and b. Imagine dividing each side of the large square, whose side is a+b, into two lengths: a and b. The a-by-a corner is a square with area a², and the b-by-b corner is a square with area b². Each of the two remaining regions has one dimension a and the other dimension b. Therefore, each remaining region is an a-by-b rectangle with area ab. Option A is correct because it identifies both dimensions and the shape. Option B is wrong because an a-by-a square has area a² and is already one of the specified squares. Option C is wrong because a b-by-b square has area b² and is also already present. Option D is wrong because a square of side a+b is the entire large figure, not one of the two remaining parts. The two rectangles together contribute 2ab, producing (a+b)²=a²+2ab+b². Memory cue: two equal crosswise pieces in the square model are ab rectangles, not additional squares.

The correct answer is A. The large square has side length a+b, so its total area is (a+b)². The four smaller regions have areas a², ab, ab, and b². Adding them gives a²+ab+ab+b²=a²+2ab+b². Therefore, the model proves (a+b)²=a²+2ab+b². Option B is wrong because the middle term there is −2ab; that identity represents a square built from a difference, not two positive ab rectangles. Option C is wrong because it represents the difference of two squares and has only two factors, not the four positive regions described. Option D is wrong because it incorrectly omits the two ab regions; in general, (a+b)² is not merely a²+b². The important step is to count both identical rectangles, so ab+ab becomes 2ab. Memory cue: a plus square has a positive doubled middle term.

The correct answer is B: 4x²+4x+1. The area of a square is side × side, so the required area is (2x+1)². Apply (a+b)²=a²+2ab+b² with a=2x and b=1. Then a²=(2x)²=4x², 2ab=2(2x)(1)=4x, and b²=1²=1. Adding these parts gives 4x²+4x+1. Option A is wrong because the square of 2x is 4x², not 2x². Option B is correct because it includes both squares and the middle rectangle areas. Option C is wrong because it omits the middle term 4x. Option D has the wrong first term, 2x², even though its middle term is 4x. A visual model would contain one 4x² square, two 2x-by-1 rectangles each of area 2x, and one unit square; together the rectangles give 4x. Memory cue: never forget the 2ab term when squaring a binomial.

The correct answer is A: t²+7t. The area of a rectangle is the product of its length and breadth. Therefore, area=t(t+7). Use the distributive property: t(t+7)=t×t+t×7=t²+7t. Option A correctly multiplies t by both terms inside the bracket. Option B, t²+7, is wrong because it adds 7 without multiplying it by t; the second product should be 7t. Option C, 7t², is wrong because there is no multiplication of 7 by t² in the expression, and it does not result from distributing t. Option D, 8t, is wrong because it treats the bracket as though t+7 were a single simple numerical combination and loses the quadratic term t². In a visual model, the t-by-t part has area t² and the t-by-7 strip has area 7t; their sum is t²+7t. Memory cue: an outside factor must multiply every term inside the parentheses.

The correct answer is A. Dividing each side into lengths a and b makes the full side a+b, so the large square has area (a+b)². Its four parts have areas a², ab, ab, and b². Adding them gives a²+ab+ab+b²=a²+2ab+b². Option A states exactly this relationship. Option B is wrong because it has a negative middle term; that would correspond to subtracting two ab regions in a difference model. Option C is wrong because it is the difference-of-squares identity and describes a different arrangement involving factors a+b and a−b. Option D is wrong because it leaves out the two ab rectangles and falsely says the sum of two square areas equals the area of the larger square. The diagram supplies two equal positive cross-regions, so their combined contribution must be +2ab. Memory cue: count regions first, then add their areas.

The correct answer is A: 8q. Divide each side of the square, q+8, into parts q and 8. In the area model, one crosswise rectangle has dimensions q and 8. Its area is q×8=8q. The second equal rectangle has the same dimensions in the opposite direction, so it also has area 8q. Option A is correct because the question asks for the area of each individual rectangle, not the combined area of both; together they would contribute 16q. Option B, 16q, is the total area of the two equal rectangles, so it is not the area of one rectangle. Option C, 64, is the area of the small 8-by-8 square. Option D, q², is the area of the q-by-q square. The complete square expansion is (q+8)²=q²+16q+64. Memory cue: one ab rectangle has area ab, while two equal rectangles together have area 2ab.

The correct answer is B: the side is z−6. Recognize the expression as a perfect-square trinomial. Using (a−b)²=a²−2ab+b², take a=z and b=6. Then (z−6)²=z²−2(z)(6)+6²=z²−12z+36, exactly the given expression. Option A, z+6, would produce (z+6)²=z²+12z+36, with a positive middle term. Option B is correct because it gives the required negative middle term and the constant 36. Option C, z−12, would produce z²−24z+144, so both the middle and final terms are wrong. Option D, z+36, would produce z²+72z+1296, which is also incorrect. In a visual model, the z² square and the 36 square are joined with two negative z-by-6 strips, giving −12z. Memory cue: half the magnitude of the middle coefficient is 6, and the negative sign gives z−6.